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Top1 Introduction
Particle swarm optimization (PSO) algorithm is a stochastic optimization technique based on the behavior of swarm (Agrafiotis and Cedeno, 2002). Stochastic search algorithms are better suited for solving highly nonlinear problems as compared to deterministic algorithms. In fact, the main motive behind developing stochastic search algorithms is to solve larger problems at a faster rate while maintaining the robustness of the algorithms. The main design idea of the PSO algorithm is closely related to evolutionary algorithms and artificial life. Just like the evolutionary algorithm, PSO is also a population-based algorithm to simultaneously search large region in the solution space of the optimized objective function. It does not necessitate the use of optimized functions such as differential, derivative, and continuous; its convergence rate is fast; and the algorithm is simple and straightforward to implement through programming. Artificial life studies the artificial systems with life characteristics (Liu, 2015). It has been successfully applied to many problems such as artificial neural network training, function optimization, fuzzy control, and pattern classification (Gong et.al., 2017 and Xue et.al., 2019), to name a few. Because of its ease of implementation and fast convergence to acceptable solutions, PSO has received broad attention (Gong et.al., 2017). Since 1995, different aspects of the original or basic version of PSO have been modified and many variants have been proposed. In PSO, particles can update their positions and velocities according to the environment change, namely it meets the requirements of proximity (the swarm should be able to carry out simple space and time computations) and quality (the swarm should be able to sense the quality change in the environment and the response). The convergence and stochastic stability study of a number of PSO variants, differing from the classical PSO in the statistical distribution of the three PSO parameters: inertia weight, local and global acceleration factors [30]. Besides the robust variant of cuckoo search (CS) algorithm, two additional optimization algorithms, i.e., GA and a modified PSO in the form of repulsive PSO with local search and chaotic perturbation (RPSOLC) are also employed for solving the considered optimization problem. The relative optimization performance of these three algorithms is also evaluated. After validating the numerical optimization procedures of the algorithms under consideration, numerous test problems available in the literature having different boundary conditions, skew angles, and aspect ratios are finally solved, and the derived solutions are reported (Kalita et.al., 2021). They presented an analytical presentation for the top limit of the particle trajectories' second-order stability areas (the so-called USL curves), which is available for most PSO algorithms. Numerical experiments revealed that adjusting the PSO parameters near to the USL curve yielded the greatest algorithm performance. Although a few review articles on PSO and its convergence analysis (see section 2 for details) have been published already (Banks et.al., 2008 and Poli et.al., 2007b) an important reason for an additional review paper is that the latest comprehensive review paper on convergence analysis of PSO have been published since then. As the latest comprehensive review paper solely on convergence analysis of PSO was published in 2013(Dong ping Tian, 2013). The need for a new review paper seems justified.
The main aim of this survey is to review the presented ideas, categorize and link most recent high-quality studies, and provide a vision for directions that might be valuable for future research. However, we briefly discuss some of these methods from theoretical perspectives: convergence to local optima, transformation invariance, and the time complexity of the methods.